Many of the families of the 67 British victims who died in the 9/11 attacks have attended services in London and New York.
Members of around 30 families who lost loved ones in the twin towers took part in a ceremony at the American Embassy in central London, while members of another 10 families took part in the service at Ground zero.
Speaking at the service outside the US Embassy, the Prince of Wales said the families had to endure an "eternity of continuing, awful, agony".
"None of us will ever forget where we were or what we were doing when on that otherwise ordinary day, and out of a clear blue sky, came so much premeditated death and destruction on scale and in a way that shocked the entire world," he said.
"At the heart of those endless and rather impersonal news reports lay the shattered lives and hopes of all those who we join here today both in London and New York."
Prince Charles said that while it was hard, seeking revenge for the attacks would never achieve peace in the end.
"It is surely only by avoiding vengefulness that we can rebuild what has been lost and save it from being lost again," he said.
Ceremonies of remembrance also took place at Westminster Abbey and St Paul's Cathedral, while Mayor Boris Johnson unveiled a sculpted memorial, made from steel recovered from Ground Zero in Battersea Park, west London.
Apart from the US, Britain lost the biggest number of casualties on 9/11.
Louis B. Susman, the American ambassador to the UK, paid tribute to the friendship between the United States and the United Kingdom.
"That unique relationship was never more evident than in the days and weeks after September 11," he said. "It was in this square, UK citizens, shocked and bewildered, but also defiant, came to show their solidarity and to pay their respects."
The service was also attended by Prime Minister David Cameron, Deputy Prime Minister Nick Clegg, Boris Johnson and Labour leader Ed Miliband.
There were some disturbances during the ceremony as anti-American protesters burned an American flag and chanted slogans such as "Obama, Obama, terrorist".
At the Ground Zero memorial site in the New York President Obama read from Psalm 46:4. "God is our refuge and strength; a very present help in trouble. Therefore will not we fear," he said.
Obama was joined by former president George W. Bush, who reminded the audience of the words of another former president, Abraham Lincoln.
"President Lincoln not only understood the heart break of his country, he also understood the cost of sacrifice, and reached out to console those in sorrow," he said.
Reading from a letter Lincoln had sent to a mother who lost five sons in the American Civil War, he said: "I pray that our heavenly Father may assuage the anguish of your bereavement, and leave you only the cherished memory of the loved and lost, and the solemn pride that must be yours to have laid so costly a sacrifice upon the altar of freedom."
Throughout the day family members and friends paid individual tributes to those lost in the attacks.
Speaking in Washington at the Pentagon, Vice President Joe Biden said attacks were "an unconscionable tragedy" and praised US military personnel.
"Never before in our history has America asked so much, over such a sustained period, of an all-volunteer fighting force. So I say without doubt or exaggeration that the 9/11 generation ranks among the greatest our nation has ever produced," he said.
In total, nearly 3,000 lives were lost in New York, Pennsylvania and Washington.
On Saturday, Obama used his weekly address to reassure the country that America "is stronger".
"We refuse to lie in fear...but we remain vigilant," he said, while paying tribute to the "military personnel and our intelligence, law enforcement and homeland security professionals", adding, "al-Qaeda is on the path to defeat".
"Across the Middle East and North Africa a new generation of citizens is showing that the future belongs to those that want to build, not destroy."
In Pennsylvania, George W Bush and Bill Clinton joined families of the passengers at the crash site of the downed Flight 93. Bush referred to those on board as "the first counter insurgency of the War on Terror". Their decision to try and overpower the hijackers led to the plane crash before it reached its target. All 40 on board perished.Suggest a correction